Integration. It's the simplest way on earth with which we can derive any formula like surface area or volume of symmetrical shapes and solids (square, circle, cube etc.). But what I've been hearing is that Archimedes figured out the basic formulas for area and volume in such an ancient age (many many years before Leibniz and Newton). I tried to have a small research on this topic as I was finding all this very interesting. So far I got these sources
So what I am yet thinking is that Calculus is just the pure mathematical form of observations (like dividing them all in $dx$) which Archimedes previously used in his age in non-matured form. However he succeeded in deriving those formula with these observations.
But can we really say that he applied the pure mathematical calculus in deriving all these equations? I rather think that it was his extraordinary observation which helped him.